Problem: Divide the following complex numbers. $ \dfrac{7+11i}{-1-4i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-1+4i}$ $ \dfrac{7+11i}{-1-4i} = \dfrac{7+11i}{-1-4i} \cdot \dfrac{{-1+4i}}{{-1+4i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(7+11i) \cdot (-1+4i)} {(-1-4i) \cdot (-1+4i)} = \dfrac{(7+11i) \cdot (-1+4i)} {(-1)^2 - (-4i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(7+11i) \cdot (-1+4i)} {(-1)^2 - (-4i)^2} = $ $ \dfrac{(7+11i) \cdot (-1+4i)} {1 + 16} = $ $ \dfrac{(7+11i) \cdot (-1+4i)} {17} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({7+11i}) \cdot ({-1+4i})} {17} = $ $ \dfrac{{7} \cdot {(-1)} + {11} \cdot {(-1) i} + {7} \cdot {4 i} + {11} \cdot {4 i^2}} {17} $ Evaluate each product of two numbers. $ \dfrac{-7 - 11i + 28i + 44 i^2} {17} $ Finally, simplify the fraction. $ \dfrac{-7 - 11i + 28i - 44} {17} = \dfrac{-51 + 17i} {17} = -3+i $